Addendum: Level spacings for integrable quantum maps in genus zero

DOI10.1007/S002200050423zbMATH Open0949.37041arXivmath-ph/0002011OpenAlexW2800282600MaRDI QIDQ1265729

Publication date: 15 December 1998

Published in: Communications in Mathematical Physics (Search for Journal in Brave)

Abstract: In this addendum we strengthen the results of math-ph/0002010 in the case of polynomial phases. We prove that Cesaro means of the pair correlation functions of certain integrable quantum maps on the 2-sphere at level N tend almost always to the Poisson (uniform limit). The quantum maps are exponentials of Hamiltonians which have the form a p(I) + b I, where I is the action, where p is a polynomial and where a,b are two real numbers. We prove that for any such family and for almost all a,b, the pair correlation tends to Poisson on average in N. The results involve Weyl estimates on exponential sums and new metric results on continued fractions. They were motivated by a comparison of the results of math-ph/0002010 with some independent results on pair correlation of fractional parts of polynomials by Rudnick-Sarnak.

Full work available at URL: https://arxiv.org/abs/math-ph/0002011

zbMATH Keywords

correlation function quantum maps statistical systems

Mathematics Subject Classification ID

Spectral problems; spectral geometry; scattering theory on manifolds (58J50) Quantum chaos (81Q50) Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests (37J35) Groups and algebras in quantum theory and relations with integrable systems (81R12) Geometric quantization (53D50) Perturbations of PDEs on manifolds; asymptotics (58J37)